True interest rates and newton's raphson method: The price of a car is $8000, and...

a6m0

New member
Joined
May 20, 2017
Messages
10
Hi,

Im having trouble with the following questions, could someone please help me out?

1. The price of a car is $8000, and an individual makes a down payment of 25% towards the purchase of the car, and secures financing for the balance over four years. If the monthly payment is $152.18, what is the true rate of interest charged by the finance company?

2. 0 = 0.001x^3+0.2x^2+0.1x-30, how would we solve for x using the newton's raphson method?

Thanks in advance ;)
 
Hi,

Im having trouble with the following questions, could someone please help me out?

1. The price of a car is $8000, and an individual makes a down payment of 25% towards the purchase of the car, and secures financing for the balance over four years. If the monthly payment is $152.18, what is the true rate of interest charged by the finance company?

2. 0 = 0.001x^3+0.2x^2+0.1x-30, how would we solve for x using the newton's raphson method?

Thanks in advance ;)
1. The price of a car is $8000, and an individual makes a down payment of 25% towards the purchase of the car, and secures financing for the balance over four years. If the monthly payment is $152.18, what is the true rate of interest charged by the finance company?

What was the total amount financed?

What equation is used to calculate the "payment" amount?


2. 0 = 0.001x^3+0.2x^2+0.1x-30, how would we solve for x using the newton's raphson method?

What are the steps of Newton_Raphson Method?

What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33

 
Hi, thanks for the reply.

I think I've got the first question sorted out, but I have a question about the second question.

My work is :

f(x) = 0 =0.001x^3+0.2x^2+0.1x-30

It's derivative: 0.003x^2+0.4x+0.1

Using Newton's method:
Initial guess =5

x1= 5 - (0.001(5)^3+0.2(5)^2+0.1(5)-30)/(0.003(5)^2+0.4(5)+0.1)
x1 ≈ 16.207

Repeating the process I get:

x2 ≈12.35242
x3 ≈11.69109
x4 ≈11.67116
x5 ≈11.67114
x6[FONT=arial, sans-serif] [/FONT]≈11.67114

so I assume 11.67114 is a root of the function, and my question is, is that the only root, or will there be others, and if there are others, how do we find those roots?

Thank You :p
 
Hi, thanks for the reply.

I think I've got the first question sorted out, but I have a question about the second question.

My work is :

f(x) = 0 =0.001x^3+0.2x^2+0.1x-30

It's derivative: 0.003x^2+0.4x+0.1

Using Newton's method:
Initial guess =5

x1= 5 - (0.001(5)^3+0.2(5)^2+0.1(5)-30)/(0.003(5)^2+0.4(5)+0.1)
x1 ≈ 16.207

Repeating the process I get:

x2 ≈12.35242
x3 ≈11.69109
x4 ≈11.67116
x5 ≈11.67114
x6≈11.67114

so I assume 11.67114 is a root of the function, and my question is, is that the only root, or will there be others, and if there are others, how do we find those roots?

Thank You :p

This is where the Fundamental Theorem of Algebrahttps://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html comes in handy. It says that any nth degree polynomial has exactly n roots (aka "solutions" or "zeros"). Your polynomial here is a third degree one, so there will be three roots. These roots may or may not be real and one or more of them may be the same as another root. The Complex Conjugate Root Theoremhttp://www.mathwords.com/c/conjugate_pair_thm.htm says that complex ("imaginary") roots always come in pairs, meaning a third degree polynomial will either have one real root and two complex roots, or three real roots. A quick visual inspection of the graph tells you that there are, in fact, three real roots.

The Newton-Raphson method seemingly only returns one root, but as your initial guess gets closer to one of the actual roots, which root it converges to may change. For instance, with this polynomial starting with any x0 >= -1, it converges to the root at approximately x = 11.671. But if you start with x0 = -2, it converges to a different root. Using the graph as a reference (you might try Desmos online graphing calculatorhttps://www.desmos.com/calculator. I use it all the time and it is amazing), can you find an initial x0 that will converge to the third root?
 
This is where the Fundamental Theorem of Algebra comes in handy. It says that any nth degree polynomial has exactly n roots (aka "solutions" or "zeros"). Your polynomial here is a third degree one, so there will be three roots. These roots may or may not be real and one or more of them may be the same as another root. The Complex Conjugate Root Theorem says that complex ("imaginary") roots always come in pairs, meaning a third degree polynomial will either have one real root and two complex roots, or three real roots. A quick visual inspection of the graph tells you that there are, in fact, three real roots.

The Newton-Raphson method seemingly only returns one root, but as your initial guess gets closer to one of the actual roots, which root it converges to may change. For instance, with this polynomial starting with any x0 >= -1, it converges to the root at approximately x = 11.671. But if you start with x0 = -2, it converges to a different root. Using the graph as a reference (you might try Desmos online graphing calculator. I use it all the time and it is amazing), can you find an initial x0 that will converge to the third root?


That was really informative, Thank You :D
 
Top